Find the value of \(\frac{{\left( {\tan \frac{\pi }{4} - \cot \frac{{2\pi }}{9}} \right)}}{{\left( {\tan \frac{\pi }{4} + \cot \frac{{2\pi }}{9}} \right)}} - \frac{{\left( {\cot \frac{{5\pi }}{{18}} - 1} \right)}}{{\left( {\cot \frac{{5\pi }}{{18}} + 1} \right)}}\)

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